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Ergodic Markov Chains
8/14/2006
Definition
• A Markov chain is called an ergodic chain if it is possible to gofrom every state to every state (not necessarily in one move).
• Ergodic Markov chains are also called irreducible.
• A Markov chain is called a regular chain if some power of thetransition matrix has only positive elements.
1
Example
• Let the transition matrix of a Markov chain be defined by
P =( 1 21 0 12 1 0
).
• Then this is an ergodic chain which is not regular.
2
Example: Ehrenfest Model
• We have two urns that, between them, contain four balls.
• At each step, one of the four balls is chosen at random and movedfrom the urn that it is in into the other urn.
• We choose, as states, the number of balls in the first urn.
P =
0 1 2 3 40 0 1 0 0 01 1/4 0 3/4 0 02 0 1/2 0 1/2 03 0 0 3/4 0 1/44 0 0 0 1 0
.
3
Regular Markov Chains
• Any transition matrix that has no zeros determines a regularMarkov chain.
• However, it is possible for a regular Markov chain to have a tran-sition matrix that has zeros.
• For example, recall the matrix of the Land of Oz
P =
R N S
R 1/2 1/4 1/4N 1/2 0 1/2S 1/4 1/4 1/2
.
4
Theorem. Let P be the transition matrix for a regular chain. Then,as n → ∞, the powers Pn approach a limiting matrix W with allrows the same vector w. The vector w is a strictly positive probabilityvector (i.e., the components are all positive and they sum to one).
5
Example
• For the Land of Oz example, the sixth power of the transitionmatrix P is, to three decimal places,
P6 =
R N S
R .4 .2 .4N .4 .2 .4S .4 .2 .4
.
6
Theorem. Let P be a regular transition matrix, let
W = limn→∞
Pn ,
let w be the common row of W, and let c be the column vector allof whose components are 1. Then
(a) wP = w, and any row vector v such that vP = v is a constantmultiple of w.
(b) Pc = c, and any column vector x such that Px = x is a multipleof c.
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Definition: Fixed Vectors
• A row vector w with the property wP = w is called a fixed rowvector for P.
• Similarly, a column vector x such that Px = x is called a fixedcolumn vector for P.
8
Example
Find the limiting vector w for the Land of Oz.
9
Example
Find the limiting vector w for the Land of Oz.
w1 + w2 + w3 = 1
and
( w1 w2 w3 )
1/2 1/4 1/41/2 0 1/21/4 1/4 1/2
= ( w1 w2 w3 ) .
9
Example
Find the limiting vector w for the Land of Oz.
w1 + w2 + w3 = 1
and
( w1 w2 w3 )
1/2 1/4 1/41/2 0 1/21/4 1/4 1/2
= ( w1 w2 w3 ) .
9
w1 + w2 + w3 = 1 ,
(1/2)w1 + (1/2)w2 + (1/4)w3 = w1 ,
(1/4)w1 + (1/4)w3 = w2 ,
(1/4)w1 + (1/2)w2 + (1/2)w3 = w3 .
10
w1 + w2 + w3 = 1 ,
(1/2)w1 + (1/2)w2 + (1/4)w3 = w1 ,
(1/4)w1 + (1/4)w3 = w2 ,
(1/4)w1 + (1/2)w2 + (1/2)w3 = w3 .
The solution isw = ( .4 .2 .4 ) ,
10
Another method
• Assume that the value at a particular state, say state one, is 1,and then use all but one of the linear equations from wP = w.
• This set of equations will have a unique solution and we can obtainw from this solution by dividing each of its entries by their sum togive the probability vector w.
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Example (cont’d)
• Set w1 = 1, and then solve the first and second linear equationsfrom wP = w.
(1/2) + (1/2)w2 + (1/4)w3 = 1 ,
(1/4) + (1/4)w3 = w2 .
• We obtain
( w1 w2 w3 ) = ( 1 1/2 1 ) .
12
Equilibrium
• Suppose that our starting vector picks state si as a starting statewith probability wi, for all i.
• Then the probability of being in the various states after n steps isgiven by wPn = w, and is the same on all steps.
• This method of starting provides us with a process that is called“stationary.”
13
Ergodic Markov Chains
Theorem. For an ergodic Markov chain, there is a unique proba-bility vector w such that wP = w and w is strictly positive. Any rowvector such that vP = v is a multiple of w. Any column vector xsuch that Px = x is a constant vector.
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The Ergodic Theorem
Theorem. Let P be the transition matrix for an ergodic chain. LetAn be the matrix defined by
An =I + P + P2 + . . . + Pn
n + 1.
Then An → W, where W is a matrix all of whose rows are equal tothe unique fixed probability vector w for P.
15
Exercises
Which of the following matrices are transition matrices for regularMarkov chains?
1. P =(
.5 .51 0
).
2. P =
1/3 0 2/30 1 00 1/5 4/5
.
3. P =(
0 11 0
).
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Exercises ...
Consider the Markov chain with general 2× 2 transition matrix
P =(
1− a ab 1− b
).
1. Under what conditions is P absorbing?
2. Under what conditions is P ergodic but not regular?
3. Under what conditions is P regular?
17
Exercises ...
Find the fixed probability vector w for the matrices in the previousexercise that are ergodic.
18
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